Isospin-breaking effects in the extraction of isoscalar and isovector spectral functions from e⁺e⁻→ hadrons

PUBLISHED VERSION Maltman, Kim

© 1998 American Physical Society

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15th April 2013

Isospin-breaking effects in the extraction of isoscalar and isovector spectral functions from e

¹

e

²

˜ hadrons

Kim Maltman

Department of Mathematics and Statistics, York University, 4700 Keele Street, North York, Ontario, Canada M3J 1P3 and Special Research Center for the Subatomic Structure of Matter, University of Adelaide, Australia 5005

~Received 16 October 1997; published 27 May 1998!

We investigate the problem of the extraction of the isovector and isoscalar spectral functions from data on e¹e²→ hadrons, in the presence of non-zero isospin breaking. It is shown that the conventional approach to extracting the isovector spectral function in ther resonance region, in which only the isoscalar contribution associated with v→pp is subtracted, fails to fully remove the effects of the isoscalar component of the electromagnetic current. The additional subtractions required to extract the pure isovector and isoscalar spectral functions are estimated using results from QCD sum rules. It is shown that the corrections are small (;2%) in the isovector case ~though relevant to precision tests of the CVC hypothesis!, but very large (;20%) in the case of thevcontribution to the isoscalar spectral function. The reason such a large effect is natural in the isoscalar channel is explained, and implications for other applications, such as the extraction of the sixth order chiral low-energy constant, Q, are discussed.@S0556-2821~98!02113-4#

PACS number~s!: 13.20.2v, 11.55.Hx, 13.40.Gp, 14.40.Cs

I. INTRODUCTION

One of the basic ingredients of the standard model is the conserved vector current~CVC!hypothesis, which postulates that the charged~isovector!weak vector current and the neu- tral isovector component of the electromagnetic ~EM! cur- rent are members of the same isovector multiplet. Since the charged current spectral function is now measured rather ac- curately int^decay@1–3#, the CVC hypothesis can be tested experimentally, provided, that is, one can extract the isovec- tor spectral function from data on e¹e²→hadrons@4–8#. In the absence of isospin breaking, this extraction is straightfor- ward since, for example, for n-pion final states, a state with an even ~odd!number of pions has even~odd!G-parity and hence can be produced only through the isovector~isoscalar! component of the hadronic EM current.

Isospin breaking, however, complicates the extraction of both the isovector and isoscalar spectral functions. Before proceeding, it is useful to clarify our notation. We define the standard SU(3)_F octet of vector currents by J_m^a 5¯qgm(l^a/2)q, wherel^a are the usual Gell-Mann matrices.

The electromagnetic current is then J_m^{E M}5J_m³1J_m⁸/

A

^{3, while}

the scalar correlators P^ab(q²) ~where we will restrict our attention to a,b53,8) are defined by

i

E

^{d4x exp~iqx! ^0uT@Jma~x!Jnb~0!#u0&}

[~q_mq_n2q²g_mn!P^ab~q²!. ~1!

Defining the spectral functions, r^ab(q2), corresponding to the P^ab(q²) in the standard manner, r^ab(q2) 5(1/p^)ImP^ab(q²), the EM spectral function is then

r^{E M}5r³³1 2

A

^3r381

1

3r^88. ~2!

r^{E M} contains the isovector spectral function, r^{33, which is} the isospin rotated version of the corresponding charged cur- rent isovector spectral function measured intdecays. Owing to the relation

s~e¹e²→hadrons!58p²a²

s r^{E M}~s!, ~3! however, the cross section for e¹e²→hadrons directly mea- suresr^{E M, and not}r^33orr⁸⁸. If isospin were not explicitly broken, this would present no problem sincer^{38would nec-} essarily vanish and, as noted above, one could in addition identify the states contributing to the isovector~33!and isos- calar~88!spectral functions by their G-parity. Near threshold it is known, from a study of the mixed isospin vector cor- relator (ab538) to two loops in chiral perturbation theory

~ChPT! @9#, that isospin breaking effects inr^{E M are negli-} gible. However, in the region of the r ^and v resonances, isospin breaking is significant, as signalled by the interfer- ence dip in the cross section for thepp final state@10#. The conventional method @5#for making corrections for this ob- served isospin breaking, in order to extract the vector com- ponent of the EM spectral function, is to first parametrize the amplitude in terms of a sum ofr ^andv Breit-Wigner reso- nance forms ~in general one includes also contributions as- sociated with the higherr ^resonances!, and having fitted the parameters to the observed cross sections, remove thev^con- tribution to the amplitude by hand. The squared modulus of the resulting modified amplitude is then used in place of the squared modulus of the original amplitude to identify that portion of the cross section to be associated with the purely isovector~33!portion of the EM spectral function.

The conventional procedure just described for correcting for isospin breaking, however, does not, in fact, produce the desired 33 component of the vector spectral function. To understand why this is the case, let us first define, for the neutral vector mesons, V5r^,v^,f, the decay constants F_V^a via

^⁰uJ_m^auV~k!&5m_VF_V^a e_m~k! ~4! whereem(k) is the vector meson polarization vector, and a 53,8. F_r⁽³⁾, F_v⁽⁸⁾ and F_f⁽⁸⁾ are nonzero in the isospin limit, while F_r⁽⁸⁾, F_v⁽³⁾ and F_f⁽³⁾ vanish in the isospin limit and hence are proportional to the isospin breaking parameter d^m5m_d2m_u. In the presence of isospin breaking, all of the neutral vector mesons, in principle, mix with one another, and so the physical states are of mixed isospin. If we con- sider theppfinal state mediated by thevexchange, then, to leading order in isospin breaking, this transition is indeed mediated by the isoscalar component of the EM current~the intermediate v contribution generated by the isovector cur- rent is second order in isospin breaking, one factor from the coupling F_v⁽³⁾, and one from the isospin violating v

→p¹p²decay vertex!, and hence should be removed if one wishes to extract only the isovector contributions. The re- maining r exchange contributions are, for a similar reason, however, not purely isovector. The reason is that if one con- siders the r exchange contribution to the EM spectral func- tion, the 38 component is first order in isospin breaking~be- ing proportional to F_r⁽³⁾F_r⁽⁸⁾). Thus, the r contribution to e¹e²→p¹p² necessarily contains a piece first order in isospin breaking, and associated with the 38 part of the EM spectral function, which must be subtracted in order to iso- late the purely isovector 33 component. A similar argument shows that there will also be a 38 contribution to the mea- sured cross section for e¹e²→v→3p, which one would have to correct for in order to isolate the purely isoscalar 88 component ofr^{E M.}

In what follows we will discuss how to perform the cor- rections associated with first order isospin breaking contribu- tions to the vector meson decay constants. The paper is or- ganized as follows. In Sec. II we first discuss some general issues, which provide useful qualitative guidelines for the subsequent discussion. In addition we discuss an analogous case involving the neutral isoscalar and isovector axial vec- tor currents, which example serves to illustrate the basic fea- tures we will meet in the vector current case of interest, but in a context where, unlike the vector case, the isospin break- ing decay constants are already known, being fixed by a next-to-leading order analysis using chiral perturbation theory. In Sec. III, we then show how, and with what accu- racy, existing QCD sum rule analyses of the mixed isospin

~38!vector current correlator can be used to extract the isos- pin breaking decay constants F_r⁽⁸⁾, F_v⁽³⁾and F_f⁽³⁾. In Sec. IV, we use the results of this analysis to evaluate the corrections required to extract the pure isovector contribution associated with the r, and pure isoscalar contributions associated with the v ^andf from the EM cross-section data, and comment on the effect such corrections would have, for example, on precision tests of the CVC hypothesis and the extraction of chiral low-energy constants ~LEC’s! via the inverse chiral sum rules method@11#.

II. SOME GENERALITIES AND AN ILLUSTRATIVE EXAMPLE

In the Introduction we have explained the reason for the existence of previously neglected isospin breaking correc-

tions in the extraction of the isoscalar and isovector spectral functions from e¹e²→hadrons data. We have, however, not yet demonstrated that such corrections can be expected to be numerically significant. As will be seen below, making nu- merical estimates for the size of these corrections is non- trivial, and we will be forced to rely on a QCD sum rule analysis of the mixed-isospin vector current correlator to make these estimates. Since it can be difficult to estimate, in a quantitative manner, the errors present in such an analysis, it is useful to consider any qualitative constraints, based on general principles, which one might use to judge the plausi- bility of the resulting solutions. We will discuss below the existence of such constraints based on the framework of ef- fective chiral Lagrangians. We will also consider, as an il- lustrative example, an analogous case in which the vector currents of the problem at hand are replaced by axial vector currents. The advantage of this example is that the isospin breaking pseudoscalar decay constants ~analogous to the isospin breaking vector meson decay constants to be deter- mined by the QCD sum rule analysis below! can, in this case, be computed with good accuracy using the methods of ChPT. This allows us to show explicitly, in a context where the numerical accuracy of the evaluation of the isospin breaking corrections is not open to question, that such iso- spin breaking corrections can play a significant role in the correct extraction of flavor-diagonal spectral contributions from the spectral functions of correlators involving mixed- flavor currents. Certain features of the relation of the relative signs and magnitudes of the corrections in the isovector and isoscalar cases, which recur in the vector channel, will also be exposed, again in a context where the accuracy of the numerical estimates is not open to question. From this ex- ample we will be able to unambiguously conclude that iso- spin breaking corrections of the type also present in the vec- tor channel must be expected to be numerically significant, especially for observables related to differences of weighted integrals of the isovector and isoscalar spectral functions, for which there will be cancellation between flavor-conserving contributions, but coherence between the isospin breaking corrections.

Let us turn then to the qualitative guidance offered, in the vector channel, by the framework of effective chiral Lagrangians. Certain qualitative features of mixing in the vector meson sector follow immediately from the properties of an effective chiral Lagrangian such as would be obtained by the Callan-Coleman-Wess-Zumino @12,13# construction

@14,15#. Note that the lowest order term in quark masses and derivatives which produces isospin mixing in the vector me- son propagator matrix involves a single power of the quark mass matrix@where, as per the usual chiral counting, external momenta are counted asO(q) and quark masses asO(q²)#. Moreover, at leading order, the isospin violating decay con- stants associated with the isospin pure states vanish. As a result, at this order, the transformation from the original iso- spin pure basis to the physical, mixed isospin basis is a ro- tation, and the isospin breaking decay constants of the physi- cal particles result purely from the mixing in the physical states. If we concentrate on r^-v mixing, this would mean that F_r⁽⁸⁾ and F_v⁽³⁾ were equal in magnitude and opposite in

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sign. If we consider effective isospin breaking operators higher order in the quark-mass–derivative expansion, new effects come into play. First, one finds non-vanishing ‘‘di- rect’’ isospin violating couplings of the external vector cur- rents to the isospin pure states from terms involving both derivatives and one power of the quark mass matrix. Second, terms involving both derivatives and one power of the quark mass matrix can produce off-diagonal mixing elements in the wave function renormalization matrix, a consequence of which is that the transformation from the isospin pure to physical basis is no longer a rotation, but rather the product of a symmetric matrix and a rotation. Third, terms with two powers of the mass matrix will produce modifications in the momentum-independent mixing terms in the vector meson propagator matrix. One would expect the effect of such higher order terms to be manifest in deviations from leading order relations such as F_r⁽⁸⁾52F_v⁽³⁾. The size of such devia- tions should be typical of next-to-leading order corrections in SU(3)_L3SU(3)_R, and hence might be as large as;30%.

We next turn to the illustrative example mentioned above, in which we replace the vector mesons and vector currents with pseudoscalar mesons and axial vector currents. Not only can several of the qualitative points just made be clearly illustrated in this case, but the actual numerical values of the relevant isospin breaking corrections can, for this example, be calculated to good accuracy using the techniques of ChPT, since all of the relevant decay constants are known at next-to-leading order in the chiral expansion. This is in con- trast to the case of the vector current spectral functions, where we will be forced to rely on a QCD sum rule analysis to obtain the isospin breaking decay constants. Let us, there- fore, define the axial current combination

A_m[A_m³1 1

A

^{3Am8 ~5!}

where A_m^3,8 are the 3, 8 members of the usual axial current octet. We then define the pseudoscalar decay constants, f_P^a, for P5p^0,h ^{and a}53,8, via

^⁰uA_m^auP~k!&5i f_P^a k_m. ~6!

At leading~second!order in the chiral counting the physical p^0,h basis is a pure rotation of the isospin pure octet basis p^3,p^8,

p⁰5p³1u0,p^8, h5p⁸2u0 p³ ~7! where u05

A

^3(md2m_u)/4(m_s2mˆ ), with mˆ5(m_d1m_u)/2, is the leading order mixing angle, and the isospin breaking decay constants are produced purely due to the ‘‘wrong- isospin’’ admixture present in the physicalp^0,h ^states,

f_p^~8!5u0F, f_h^~3!52u0F ~8! where F is a second order LEC, identical to both f_p⁽³⁾5f_p, the physical pion decay constant, and f_h⁽⁸⁾5f_h, the physical h decay constant, at this order in the chiral expansion. When one considers the full effective Lagrangian at next-to-leading order, however, one finds that ~1!there is an infinite renor-

malization required to regularize the mixing ofp^3,p^{8 even} if one works with the versions of the fields renormalized in the isospin limit,~2!there is indeed an off-diagonal element produced in the wave function renormalization matrix and

~3!there are indeed ‘‘direct’’ isospin breaking meson-current vertices. ~For a detailed exposition of the above features be- yond leading order see Ref. @16#.! The effect of all these features is to produce the following results for the isospin breaking decay constants@17#, written in terms of the next- to-leading order expressions for the p^{0 and} h ^{decay con-} stants, which differ at this order in the chiral expansion:

f_p^~8!5e1f_p

f_h^~3!52e2f_h. ~9! In Eq.~9!, the quantitiese1,2differ by terms which are next- to-leading order in the chiral expansion ~the complete ex- pressions can be found in Ref.@17#!and, using the observed experimental ratio of f_K/ f_p, the next-to-leading order ex- pressions for the isospin conserving decay constants imply f_h/ f_p.1.3@17#.

We are now in a position to consider the analogue of the vector current case of interest. To this end we imagine that we would like to obtain the p^{0 and} h contributions to the isovector and isoscalar axial spectral functions of the scalar correlators,P1

33(q²) andP1

88(q²), defined by

P_mn^ab5i

E

^{d4x eiqx^0uT@Ama~x!Anb~0!#u0&}

[P1

ab~q²!q_mq_n1P2

ab~q²!~q²g_mn2q_mq_n!. ~10!

~We concentrate on the scalar correlators, P1

ab, since it is these correlators which contain the pole contributions analo- gous to those of the vector mesons in the vector current correlators.! It is straightforward, in this case, to simply evaluate the contributions to the spectral functions, r1

ab. Keeping only terms up to first order in isospin breaking and to next-to-leading order in the chiral expansion, one finds that

r1

33~q²!5@f_p^~3!#² d~q²2m_p²! r1

38~q²!5f^~_p^3!f_p^~8! d~q²2m_p²!1f_h^~3!f_h^~8! d~q²2m_h²! r1

88~q²!5@f_h^~8!#² d~q²2m_h²!. ~11! Let us imagine, however, that the way we had to go about extracting these contributions was by analyzing the ‘‘experi- mental’’ spectral function r1

A(q²) of the scalar correlator, P1

A, appearing in the analogue,P_mn^A , of the correlator of the product of two EM currents, i.e.,

P_mn^A 5i

E

^{d4x eiqx^0uT@Am~x!An~0!#u0&}

[P1

A~q²!q_mq_n1P2

A~q²!~q²g_mn2q_mq_n!. ~12! The analogue of the standard vector current extraction of the isovector component would then consist of identifying the

p⁰pole contribution with the isovectorr1

33component ofr1 A

and the h pole term with the isoscalar r1

88 component thereof. We can see, however, that this identification is in- correct. Indeed, the p⁰ contribution to r1

A, to first order in isospin breaking, is straightforwardly found to be, not r1

33, but

@r1 A#_p5r1

331 2

A

^3@r138#p

5

S

^@f~p3!#21

A

^23fp~3!f~p8!

D

^{d~q22mp2!. ~13!}

Similarly, the h pole contributions to r1

A, to first order in isospin breaking, is not ¹₃r1

88 but

@r1 A#_h51

3r1 881 2

A

^3@r138#h

5

S

^13@fh~8!#21

A

^23f~h3!fh~8!

D

^{d~q22mh2!. ~14!}

Thus, to ‘‘extract’’ the isovector and isoscalar spectral func- tions from the ‘‘experimental’’ spectral function one would actually have to know the isospin violating decay constants and then form the combinations

r1 335@r1

A#_p2 2

A

^{3f~p3!fp~8! d~q22mp2!}

r1 8853@r1

A#_h22

A

^{3 f}_h^~3!f_h^~8! d~q²2m_h²!.

~15! Note that, because of the structure of the original current, the size of the isospin breaking correction in the isoscalar case is naturally larger by a factor of 3 than that in the isovector case. If we work with the values of e1,2 from Ref. @17#

@which correspond to the value of the isospin breaking mass ratio r5(m_d2m_u)/(m_d1m_u) used in the sum rule analysis we will employ below#,

e151.37310²², e251.11310²², ~16! we find that that the ‘‘experimental’’ pion pole contribution to r1

A is a factor of 112e1/

A

³51.016 larger than the true isovector spectral function, and that the nominal ‘‘experi- mental’’ isoscalar spectral function, obtained by taking theh pole contribution and multiplying by 3, is smaller than the true isoscalar spectral function by a factor of 122

A

³e2

50.961. Thus, to extract the true isovector and isoscalar spectral functions, one would have to multiply the nominal

‘‘experimental’’ ones by 0.984 and 1.040, respectively. Note that the corrections go in the opposite direction for the two cases and that the magnitude of the correction is significantly larger in the isoscalar case. The reason for the latter feature of the results has already been explained. The reason for the former is that the leading order result that the p^{0 and} h contributions to the 38 spectral function differ in sign, but

not in magnitude, is still approximately satisfied by the next- to-leading order expressions, as expected based on the gen- eral arguments above. Note also that the effect of these cor- rections can be greatly enhanced if one considers combinations which would vanish at leading order in the chiral counting. As an example, if we consider the integral over the 88-33 spectral function, the nominal value~without the above corrections!is 0.69f_p², while the actual value~af- ter corrections! is 0.77f_p². The isospin breaking corrections thus represent a 12% increase for this quantity, much larger than one would guess based on the typical few percent size of familiar isospin breaking corrections.

We will see in what follows that many of the features of the axial current example are recapitulated in the vector cur- rent case. In fact, because in the limit in which one considers the vector meson multiplet to be ideally mixed, but takes the decay constants to be otherwise determined by SU(3)_F sym- metry, thevEM decay constant is a factor of 3 smaller than the r EM decay constant~rather than just the factor of

A

³

difference between the ‘‘A’’ decay constants of thep^{0 and} the h in the above example!, the discrepancy between the size of the corrections in the isovector and isoscalar channels would be expected to be even greater in the vector case. We will see below that this expectation is indeed borne out.

III. QCD SUM RULE EXTRACTION OF THE ISOSPIN BREAKING VECTOR MESON DECAY CONSTANTS Since the 3 and 8 components of the vector current octet occur in the standard model only in the combination J_m^{E M}, it is not possible to directly determine the isospin breaking vector meson deay constants experimentally. One can, how- ever, obtain indirect access to these quantities via a QCD sum rule analysis of the vector current correlator P³⁸.

The basic idea of such a QCD sum rule analysis@18–20# is straightfoward. From the behavior ofP³⁸(q²) as q²→`in QCD, it is known that P³⁸ satisfies an unsubtracted disper- sion relation. This dispersion relation allows one to relate the dispersion integral over hadronic spectral functionr^{38to the} value ofP³⁸at large spacelike value of q²52Q². The latter can be expressed in terms of vacuum condensates using the operator product expansion ~OPE!, while the hadronic spec- tral function depends on the isospin breaking and isospin conserving decay constants of the various vector mesons.

The utility of this relation is greatly enhanced, as first noted in Ref.@18#, if the original dispersion relation is Borel trans- formed since, in that case, the higher s portions of the trans- formed hadronic spectral integral are exponentially sup- pressed, while the contributions of higher dimensional operators on the OPE side are simultaneously factorially sup- pressed. In favorable circumstances one is then able to write the dispersion relation in a form in which the parameters of a small number of resonances dominate the hadronic side, and the contributions of a small number of vacuum condensates of low dimension operators ~which condensates can be de- termined from other sum rule analyses! dominate the OPE side. One can then use the known values of the vacuum condensates to extract the ~in our case unknown!resonance parameters. Note that it is far preferable, in the case of inter-

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est to us, to investigate the isospin breaking decay constants by analyzing an isospin breaking correlator, rather than by looking at the isospin breaking corrections to a correlator which also has an isospin conserving piece. This is because of the fact that, in the latter case, one would have to consider isospin breaking not only in the vector meson decay con- stants, but also in the continuum thresholds, which would be required in the spectral Ansa¨tze in order to model the higher s portions of the hadronic spectral function. Since that por- tion of the spectral model is a rather crude representation of the actual continuum, this would introduce potentially large, and difficult to control, uncertainties into such an analysis.

Let us turn then to the sum rule analysis of the correlator P³⁸(q²). We will, in fact, use the results of existing analyses

@21–24#of the related correlator,P^rv(q²), defined by P_mn^rv~q!5i

E

^{d4x exp~iq•x!^0uTJmr~x!Jnv~0!u0&}

[~q_mq_n2q²g_mn! P^rv~q²!, ~17! where J_m^r5(u¯gmu2¯dgmd)/2 and J_n^v5(u¯gmu1¯dgmd)/6.

This is possible because, truncating the OPE at O(m_q), O(as), O(aE M) and operators of dimension 6

~with either the vacuum saturation hypothesis, or the rescaled version thereof, for the four-quark condensate contributions!, the s¯s portion of J_m⁸ does not contribute to the OPE repre- sentation of P³⁸(q²). One then has

@P³⁸~q²!#O PE5

A

³ @P^rv~q²!#O PE, ~18! where, after Borel transformation~indicated by the operator B) of the truncated OPE expression, one finds@21#

B@P^rv~s!#O PE51

12

F

^{c0 M21c11Mc221Mc34}

G

^{, ~19!}

with M the Borel mass,

c₀5 aE M

16p³ c₁5O~m_q²!;0

c₂54

S

^{mu22m1d~g11g!}

D

^{^q¯ q&0}

c₃5224p g

81 ask ^^{¯ qq} &0

2228p

81 aek ^^{¯ qq} &0

2, ~20!

where g[^^{¯ dd} &0/^^{¯ uu} &021, ^^{¯ qq} &0[(^^{¯ uu} &01^^{d¯ d}&0)/2, ae

and as are the electromagnetic and strong coupling con- stants, respectively, and k is the parameter describing the deviation of the four-quark condensates from their vacuum saturation values.

The hadronic spectral function,r³⁸, is parametrized as r³⁸~s!5 1

4

A

^{3@frdr~s!2fvdv~s!1ffdf~s!1•••#}

~21!

where

dV~s!51 p

m_VGV

~s2m_V²!²1m_V²GV

2 ~22!

@which reduces tod^(s2m_V²) in the narrow width approxima- tion#, and we have followed the notation and sign conven- tions employed in the literature for the parameters f_V de- scribing the strengths of the various vector meson contributions to the spectral function in question. Note that, in the notation introduced above,

f_V

4

A

^{356FV~3!FV~8! ~23!}

where the upper sign holds for V5r^,f, . . . , and the lower sign for V5v^.

A few comments are in order concerning the form of Eq.

~21!. The first concerns the presence of the f contribution, which was not included in the earliest sum rule analyses

@21,22#. Recall that we expect, as a leading order result, that f_r.f_v ~the absence of a minus sign in this relation is a consequence of the sign convention for the definition of f_v).

This means that, since from far in the spacelike region ther andv masses appear essentially the same, one must expect significant cancellation between ther ^andvcontributions to the original dispersion relation for large values of Q². This is especially true in the narrow width approximation. Based on this observation, it was realized, in Ref.@23#, that, although f_fcould be expected to be significantly smaller than f_ror f_v

~of order 6–7 % if one makes an estimate based on the Par- ticle Data Group evaluation of the deviation of vector meson mixing from ideal mixing@23#!, the contribution of thef ^to the actual sum rule need not be negligible. The sum rule was then re-analyzed, including thefcontribution, though still in the narrow width approximation. The results supported the qualitative arguments regarding the importance of including the f contribution, and simultaneously cured a physically unpleasant feature of the earlier analyses, in which contribu- tions from ther8^-v8 region of the spectral function were as important, or more important, than those from the r^-v ^re- gion in determining, through the original dispersion relation, the value of the correlator near q²50. It was, however, then subsequently pointed out @24# @Iqbal, Jin and Leinweber

~IJL!#, again because of the high degree of cancellation be- tweenr ^andv contributions to the dispersion integral in the far spacelike region, that the use of the narrow width ap- proximation for the r might also be a rather poor one. This was borne out by the numerical analysis of Ref. @24#. Of particular note is the fact that, introducing the r ^{width into} the spectral Ansatz for the r contribution, one finds that f_r and f_v are lowered in magnitude by a factor of;6, and f_f by a factor of;2 compared to the values extracted using the narrow width analysis. With the above understanding in mind, we will, therefore, in what follows, employ the spec- tral Ansatz of Eq.~21!in the form arrived at in Ref.@24#, i.e.,

with thev ^andftreated in the narrow width approximation, but the r treated using the Breit-Wigner form given in Eq.

~22!.

It should be stressed that, in using the results of Ref.@24#, the present analysis relies only on the extraction of the had- ronic spectral function for the mixed-isospin vector current correlator, accomplished in that reference using QCD sum rules, and not on the attempt by the authors of Ref. @24#to interpret this spectral function in terms of off-shell vector meson propagators. The latter interpretation~and that of Ref.

@22#!is necessarily incorrect, since off-shell Green functions are well known to be altered by redefinitions of the hadronic fields and, as such, are not capable of being related to physi- cal objects such as the mixed-isospin vector current cor- relator ~an earlier claim by the present author, contained in Ref. @23#, that the rescaled versions of the vector fields rep- resented a possible field choice for the vector mesons is also incorrect!. The only questions relevant to the use of the spec- tral function solution of Ref. @24# in the present work are, then,~1!is the sum rule reliable for the scales at which it is employed and~2!is the original Ansatz for the spectral func- tion plausible in form? While it is not possible to provide a rigorous proof of the suitability of the analysis of Ref.@24#, there is considerable indirect evidence in its favor. First, al- though a resonance-saturation Ansatz involving s-dependent widths, but constant real parts of the resonance self-energies, does not rigorously implement the constraints of analyticity and unitarity, it is well known from phenomenological stud- ies of e¹e²→hadrons that such Ansa¨tze, nonetheless, can provide fits of very high numerical accuracy to the experi- mentally measured spectral function~see Ref.@25#for a re- cent detailed discussion!. This is also true of the resonance- saturation fits to the timelike pion form factor measured in hadronict decays mediated by the isovector current~where, for example, the naive form of the resonance contributions involving s dependence only in the widths produces a fit of even slightly better quality than does an Ansatz employing the Gounaris-Sakurai form, which correctly implements ana- lyticity and unitarity—see the results of Table 3 of Ref.@1#!. Moreover, a recent analysis of the isovector ~isospin con- serving!vector correlator using a continuous family of finite energy sum rules as a means of implementing the dispersion relation implicit in QCD sum rules shows that fitting a naive form of the resonance saturation Ansatz for the hadronic spectral function to the OPE representation ~where the widths and decay constants of the resonances are free param- eters determined by the fit to the OPE! reproduces the ex- perimentally measured spectral function@1#to an accuracy of a few percent and, moreover, requires a phenomenological value of ther width within a few MeV of that obtained by direct fitting of the experimental data using the favored hid- den local symmetry ~HLS! model in Refs.@25,26#. This in- dicates that, at least in the isospin conserving case, matching of an Ansatz of the type employed in Ref. @24#to the OPE representation of the vector current correlator provides a good fit to the actual vector current spectral function. Note that the matching between the hadronic and OPE sides of the conventional QCD sum rule obtained in Ref. @24# is also considerably better than that obtained in earlier analyses us-

ing the narrow width approximation for the r, further indi- cating the necessity of employing an Ansatz of the IJL form.

The only improvement in the IJL Ansatz one might have hoped for was the use of a variabler width, to be fixed by the sum rule analysis. Since, however, in the isospin con- serving case, the result of such an exercise is to essentially reproduce the width employed by IJL, it seems unlikely that the analysis would have been significantly altered by such an extension.

In light of the discussion above, we should thus be able to extract the desired isospin breaking vector meson decay con- stants from the results of Ref.@24#. The results we use below, however, differ somewhat from those quoted in that refer- ence, and the reasons for this difference must first be ex- plained. The first point in need of clarification is related to the fact that there are two sets of results associated with the analysis using the physicalrwidth in Ref.@24#. Of these two sets, only the one contained in the column labelled ‘‘physical widths’’ and ‘‘no constraint’’ in Table 1 of that reference should be employed. The reason for rejecting the other set

~labelled ‘‘constrained’’! is that these results were obtained assumingP³⁸(m_v²) could be extracted from thev→pp^con- tribution to the physical cross section. This assumption, how- ever, as we have seen above, neglects the presence of addi- tional P³⁸ contributions in the r exchange contribution to the cross section, and hence is incorrect. The second modi- fication we make in employing the results of Ref. @24#con- cerns the magnitude of the errors quoted on the extracted values of the parameters f_V. It turns out that, in Ref.@24#, an overly conservative error was assumed on the crucial input isospin breaking light quark mass ratio, r_m5m_u/m_d. The authors of Ref. @24# employed r_m50.5060.25. The ratio, r_m, however, is actually much better constrained than this from ChPT analyses@27#. As a consequence, the quoted er- rors in Ref. @24# are unnecessarily inflated. The authors of Ref. @24# have kindly provided unpublished results corre- sponding to the more realistic input r_m50.5460.04 em- ployed in earlier sum rule analyses of the correlator at hand.

We will, therefore, employ, for determining the errors on the extraction of the desired isospin breaking vector meson de- cay constants, the results of the ‘‘unconstrained’’ fit obtained using the modified input above, r_m50.5460.04. The results, which correspond to a stable Borel regime 1.15 GeV,M,2.45 GeV, are@28#

f_r54

A

^3F_r^~3!F_r^~8!50.003060.0012 GeV² f_v524

A

^3F_v^~3!F_v^~8!50.002560.0009 GeV² f_f54

A

^3F~_f^3!F_f^~8!520.000260.0002 GeV².

~24!

The contributions from the r8 ^andv8 are found to be very small, and cannot be reliably extracted from the sum rule in its current form. Note that the reliability of the results is supported by that fact that they display two features which correspond to our general expectations: first, that f_r.f_vand, second, that f_f is of order 6–7 % of f_r and f_v.

KIM MALTMAN PHYSICAL REVIEW D 58 014008

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We are now in a position to evaluate the isospin breaking decay constants F_r⁽⁸⁾, F_v⁽³⁾and F_f⁽³⁾. To do so we employ the results of Eq.~24!, together with the relations

F_V^{E M}5F_V³1 1

A

^{3FV8, ~25!}

where the physical EM vector meson decay constants, F_V^{E M}, determined experimentally from the partial widths for the decays V→e¹e², are

F_r^{E M}515463.6 MeV F_v^{E M}545.960.8 MeV

F_f^{E M}5279.162.3 MeV. ~26! The sign of thefdecay constant in Eq.~26!has been chosen to be consistent with expectations from SU(3)_F symmetry and ideal mixing. Note that the relative magnitudes in Eqs.

~26!are roughly in line with the expectations of that limit~in which one would expect ther^,v ^andf decay constants to be in the ratios 3:1:2

A

2). It is then straightforward to solve for F_V⁽³⁾and F_V⁽⁸⁾separately. One finds, for the isospin break- ing decay constants,

F_r^~8!52.861.1 MeV F_v^~3!524.261.5 MeV

F_f^~3!50.2160.21 MeV, ~27! where the quoted errors are totally dominated by the errors of the sum rule fit values of the products of decay constants.

The values of the isospin conserving decay constants then follow immediately from Eqs. ~25!, ~26!and~27!. We will quote them in the form of ratios to the relevant experimental values, which form allows one to directly compute the addi- tional corrections required to obtain the pure isovector and isoscalar spectral functions from those obtained convention- ally, i.e., via the standard analysis described above. We find

F

^FFrE M^r~3!

G

^{21520.01160.0043}

F AF3Fv~8vE M! G2150.09160.029

F AF3Ff~8fE M! G2150.002760.0027. ~28!

IV. CONSEQUENCES FOR THE ISOVECTOR AND ISOSCALAR SPECTRAL FUNCTIONS If we drop terms second order in isospin breaking, then ther^,v ^andf resonance contributions to the isovector and isoscalar vector current spectral functions are easily seen to be

@r³³~q²!#V5@F_r^~3!#²dr~s!

@r⁸⁸~q²!#V5@F_v^~8!#²d_v~s!1@F_f^~8!#²df~s!.

~29! The results of the standard extractions, in contrast, are ob- tained by replacing F_r⁽³⁾ with F_r^{E M}, F_v⁽⁸⁾ with

A

^3F_v^{E M and}

F_f⁽⁸⁾with

A

^3F_f^{E M}. The corrections to be applied to the stan- dard contributions in order produce the true resonance con- tributions to the isovector and isoscalar spectral functions are then given by the ratios

F

^FFrE M^r~3!

G

^{250.97960.0086}

F A3 FFv~8!vE MG251.18960.065

F A3 FFf~8!fE MG251.005460.0054. ~30!

From the above results we see that the standard procedure leads to an overestimate of the vector spectral function by 2.160.9 %. This is still noticeably smaller than the ;5%

errors on the e¹e²→hadrons cross sections in the resonance region. As a result, it is not yet possible, when comparing to t data, to see the effect of the r³⁸ contributions to the e¹e²→hadrons spectral functions extracted using the con- ventional analysis.~See Fig. 10c of Ref.@1#for a comparison of the spectral functions as extracted fromt^{and e1e2data.}

The above correction would lower the e¹e² points by;25 nanobarns in the region of the r ^peak.!A correction of this size, however, would certainly become important if one wished to make tests of the CVC hypothesis at the 1% level.

A much greater surprise is the size of the correction re- quired in the case of thevcontribution to the isoscalar spec- tral function. While a 19% isospin breaking correction might sound unnaturally large, the size of the correction is, in fact, completely natural, and easily understood. The main features of the result follow from considering only the leading order contributions as discussed in Sec. II above. Let us, therefore, consider the approximation in which one considers only the leading @O(m_q,q⁰)# contributions to r^-v mixing, neglects the ‘‘direct’’ isospin violating contributions to the vector me- son decay constants~which are also higher order!, and works in the ideal mixing/SU(3)_F approximation in which the EM decay constant of the pure isospin 1 component of ther ^{is 3} times that of the pure isospin 0 component of thev^{. Writing} r5rI1e vI, v5vI2e rI, ~31! where the subscript I denotes the isospin pure states, ande^is O(dm), the physical EM decay constants become

F_r^{E M}5F_r^I1e^F_v^I.F_r^I

S

^113e

D

F_v^{E M}5F_v^I2e^F_r^I.F_v^I~123e!. ~32!

The fractional correction in thevcase is thus expected to be

;9 times as big as that for ther case. That the actual cor- rections turn out to be exactly a factor of 9 different is a numerical accident, but the large relative size of the correc- tions is completely natural, and associated with the smallness of the EMv coupling and the pattern of mixing in the vector meson sector. Note also the fact that the corrections are of opposite signs is exactly what one expects based on the gen- eral arguments above.

It is not just the isovector spectral function, with its rela- tion to the CVC hypothesis, for which the corrections ob- tained above are of phenomenological interest. The differ- ence of the isovector and isoscalar spectral functions also enters a number of interesting sum rules, and these sum rules must, therefore, also be corrected for the effects just dis- cussed. As an example we will consider the extraction of the sixth ~chiral!order low energy constant Q(m⁾ ~in the nota- tion of Refs.@29,9#!from the inverse moment chiral sum rule

@29#

E

4m_p²

`

ds~r³³2r⁸⁸!~s!

s 516~m_K²2m_p²! 3F² Q~m²! 1 1

48p^2log

S

^mmK2p2

D

1

S

^{L9r~m22!1p2LF10r2~m2!}

D

3

F

^mp2log

S

^mmp22

D

^2mK2log

S

^mmK22

D G

^.

~33! In Eq. ~33!, L_k^r are the scale-dependent renormalized fourth order LEC’s of Gasser and Letuwyler @17#, and m ^{is the} ChPT renormalization scale. The form of this equation relies on the two-loop expressions for the 33 and 88 correlators obtained in Ref.@11#. The difference of the 33 and 88 spec- tral functions also enters a method of determining the strange current quark mass originally suggested by Narison@30#. In this application, a weighted integral overr^33(q2)2r^{88(q2) is} performed, the weight function being that which enters in- clusivetdecays. The corrections in this case, and the result- ing values of the strange quark mass, will be treated in a separate paper @31#.

In the analysis of the sum rule, Eq. ~33!, performed in Ref. @29#, ther contribution to the spectral integral was ob- tained from t decay data, and hence does not require the correction discussed above. Thev^andfcontributions, how- ever, are determined from the experimental V→e¹e²partial widths, and hence contain contributions fromr^{38which must} be removed. The uncorrectedr^,v^andfcontributions to the spectral integral are@29#0.0374,20.0103 and20.0204, re- spectively. Implementing the corrections above, one finds that the sum of these three contributions is reduced from

0.0067 to 0.004660.0007, a downward shift of 31%. Includ- ing the estimates for the 4p ^{and K¯ K}pp contributions as evaluated in Ref. @29#, we find that the central value for Q(m_r²) is shifted from 3.7310²⁵ to 2.2310²⁵, a change of 41%. As noted above, because of the cancellations inherent in forming r^33(q2)2r^88(q2) @the combination vanishes in the SU(3)_F limit#, the effect of the isospin breaking correc- tions is large. A similar effect is found in the case of the strange quark mass analysis.

V. SUMMARY

We have shown that contributions to e¹e²→hadrons in- volving an intermediate state r or intermediate statev ^or f contain contributions from the isospin violating 38 vector spectral function which are not negligible, and must be re- moved if one wishes to extract the isovector 33 and isoscalar 88 spectral functions from e¹e²→hadrons data. Using the results of a QCD sum rule analysis of the 38 correlator, we have been able to estimate the isospin violating vector meson decay constants required to make these subtractions. We find that the isovector spectral function is ;2% smaller than what one would obtain by assuming it was identical to the full experimentalr contribution, and that thev contribution to the isoscalar spectral function is ;19% larger than what one would obtain from experiment without making this cor- rection. We have also explained why it is unavoidable that

~1!the isoscalar correction will be much larger than the is- ovector correction ~by roughly an order of magnitude!, and

~2!the sign of ther ^andv corrections in the isovector and isoscalar cases, respectively, will be opposite. A conse- quence of the second point is that all observables related to weighted integrals over the difference of the 33 and 88 spec- tral functions will receive large isospin breaking corrections, dominated by those which need to be made to correctly ob- tain thev contribution to the 88 isoscalar term.

Finally, we note that it might be possible to reduce the errors on the extractions of the isospin breaking decay con- stants by updating the sum rule analysis of the 38 correlator using recent improved values for the input parameters, and evaluating higher orderas corrections to the Wilson coeffi- cients appearing in the D52,4 terms of the OPE. This will be the subject of future investigations.

ACKNOWLEDGMENTS

The author would like to thank Andreas Ho¨cher of the ALEPH Collaboration for elaborating on the procedure used in making isospin breaking corrections to e¹e²→hadrons data in the CVC hypothesis test of Ref.@1#, and Derek Lein- weber for making available the unpublished results em- ployed in Sec. III above. The ongoing support of the Natural Sciences and Engineering Research Council of Canada and the hospitality of the Special Research Centre for the Sub- atomic Structure of Matter at the University of Adelaide, where this work was performed, are also gratefully acknowl- edged.

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